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Preconditioning of a pollution-free discretization of the Helmholtz equation

Harald Monsuur

TL;DR

This work introduces a pollution-free first-order system least squares (FOSLS) formulation for the Helmholtz equation and solves the resulting saddle-point system with a block preconditioner that uses a Schur-complement on L_2 and a Hermitian SPD test-space preconditioner built via subspace corrections. The method achieves quasi-best (pollution-free) approximations under suitable test-space enrichment and provides a practical, residual-based a posteriori error estimator that enables adaptive refinement. Numerical experiments on plane-wave and scattering problems demonstrate linear growth of MINRES iterations with the wave number κ and show the effectiveness of adaptive refinement and an algebraic-error stopping criterion. The approach is easy to implement, applicable to general domains including scattering setups, and offers O(n) memory with robust performance across mesh refinements and wave numbers, aided by a carefully designed multilevel preconditioner and smoother strategy.

Abstract

We present a pollution-free first order system least squares (FOSLS) formulation for the Helmholtz equation, solved iteratively using a block preconditioner. This preconditioner consists of two components: one for the Schur complement, which corresponds to a preconditioner on $L_2(Ω)$, and another defined on the test space, which we ensure remains Hermitian positive definite using subspace correction techniques. The proposed method is easy to implement and is directly applicable to general domains, including scattering problems. Numerical experiments demonstrate a linear dependence of the number of MINRES iterations on the wave number $κ$. We also introduce an approach to estimate algebraic errors which prevents unnecessary iterations.

Preconditioning of a pollution-free discretization of the Helmholtz equation

TL;DR

This work introduces a pollution-free first-order system least squares (FOSLS) formulation for the Helmholtz equation and solves the resulting saddle-point system with a block preconditioner that uses a Schur-complement on L_2 and a Hermitian SPD test-space preconditioner built via subspace corrections. The method achieves quasi-best (pollution-free) approximations under suitable test-space enrichment and provides a practical, residual-based a posteriori error estimator that enables adaptive refinement. Numerical experiments on plane-wave and scattering problems demonstrate linear growth of MINRES iterations with the wave number κ and show the effectiveness of adaptive refinement and an algebraic-error stopping criterion. The approach is easy to implement, applicable to general domains including scattering setups, and offers O(n) memory with robust performance across mesh refinements and wave numbers, aided by a carefully designed multilevel preconditioner and smoother strategy.

Abstract

We present a pollution-free first order system least squares (FOSLS) formulation for the Helmholtz equation, solved iteratively using a block preconditioner. This preconditioner consists of two components: one for the Schur complement, which corresponds to a preconditioner on , and another defined on the test space, which we ensure remains Hermitian positive definite using subspace correction techniques. The proposed method is easy to implement and is directly applicable to general domains, including scattering problems. Numerical experiments demonstrate a linear dependence of the number of MINRES iterations on the wave number . We also introduce an approach to estimate algebraic errors which prevents unnecessary iterations.
Paper Structure (36 sections, 5 theorems, 53 equations, 14 figures)

This paper contains 36 sections, 5 theorems, 53 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The Helmholtz equation
  3. Numerical approximation of the Helmholtz equation
  4. Ultra-weak first order system formulation, and optimal test norm.
  5. 'Practical' method
  6. Solving the algebraic system
  7. Adaptivity and error estimation
  8. Organisation
  9. Notations
  10. Helmholtz equation
  11. Ultra-weak first order formulation
  12. Pollution-free least squares approximations
  13. 'Practical' method
  14. Euler-Lagrange equations
  15. A posterior error estimation and boosted approximation.
  16. ...and 21 more sections

Key Result

Theorem 2.1

For both being Hilbert spaces equipped with their canonical norms $\|\cdot\|_U:=\|\cdot\|_{L_2(\Omega) \times L_2(\Omega)^d}$ and $\|\cdot\|_V:=\|\cdot\|_{H^1(\Omega) \times H(\mathop{\mathrm{div}}\nolimits;\Omega)}$, it holds that $B_\kappa \in \mathcal{L}\mathrm{is}(U,V')$.

Figures (14)

  • Figure 1: Left: the non-trapping domain with its initial triangulation. Right: the trapping domain with its initial triangulation.
  • Figure 2: Pollution factors for the example from Section \ref{['chap4:example1']}, for different values of $\kappa$ and $p=3$. Left: $\tilde{p}=5$, right: $\tilde{p}=6$
  • Figure 3: Pollution factors for the example from Section \ref{['chap4:example2']}, for different values of $\kappa$ and $p=3$. Left: $\tilde{p}=5$, right: $\tilde{p}=6$
  • Figure 4: Pollution factors for the example from Section \ref{['chap4:example3']}, for different values of $\kappa$ and $p=3$. Left: $\tilde{p}=5$, right: $\tilde{p}=6$
  • Figure 5: Condition number of the preconditioned system for the example from Section \ref{['chap4:example1']}, for different values of $\kappa$. Left: $p=3$ and $\tilde{p}=5$, right: $p=3$ and $\tilde{p}=6$.
  • ...and 9 more figures

Theorems & Definitions (14)

  • Theorem 2.1
  • remark 2.2
  • Theorem 2.3
  • remark 2.4
  • Theorem 2.5: 204.18
  • remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more